A characterization of type $p$ and cotype $p$ separable Banach spaces is given in terms of integrability properties of Levy measures. The following consequences are derived: (i) a separable Banach space is isomorphic to Hilbert space if and only if the set of Levy measures on it coincides with the set of Borel measures which integrate the function $\min (1, \|x\|^2)$; and (ii) the classical Levy-Khintchine representation of characteristic functions of infinitely divisible distributions holds in separable Banach spaces of cotype 2, in particular, in the separable $L_p$ spaces for $p \in \lbrack 1,2\rbrack$.
"Type, Cotype and Levy Measures in Banach Spaces." Ann. Probab. 6 (4) 637 - 643, August, 1978. https://doi.org/10.1214/aop/1176995483